The first session is the assessment of the problem. You will start out discussing whatever the problem is. It feels sort of like talking to a doctor about a medical problem. He/she will ask if you are experiencing different symptoms. If so, you will be ask to give an example. The therapist will take a lot of notes on your symptoms. After the diagnostic portion of the session, there will be more of a get to know you section. In order to provide good therapy, one must know a little background about the client. This will probably include your cultural background, religious affiliation if any, sexual orientation, family and friends background and just basic knowledge about you.
The second session is probably when you will be ask what you want out of therapy. You will set up some treatment goals to work on.
At any time, you can ask the therapist questions to see what his/her background in working with your issues are. I suspect that most therapists have good experience in working with depression and anxiety related issues because of how common the problem is. However, if you have some other issue like a preference for a specific gender, cultural group, religious affiliation or experience in working with people with childhood abuse. It would be a good idea to make these preferences known.
After that, the therapist will try to move at your pace. But, you will probably find that talking to him/her will be like being on the wrong end of a blunt object at times. Since we are asking them to help us, he or she will have to challenge your ways of thinking at times. But, a therapist will seem to know how far he/she can go in most situations which is nice.
Good luck on your new and exciting journey in therapy and hopefully healing.